alternatively use the fact that (rather than hand waving arguments) |z|= sqrt(x**2+y**2) where z=x+iy is a complex number, x,y real.

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|z+4i| + |z-4i| = 10 means that the locus of z is the set of points each of whose sum of distances from two fixed points (4i, -4i) is a constant (=10). Is this not just the same as showing that (x,y) satisfy (x/a)^2 + (y/b)^2 = 1. I don't see how it is any less rigorous, and definitely disagree with your description of it as hand waving. tell me where I'm wrong.

when you get round to demonstrating that circles and straight lines are sent to circles and straight lines under mobius transformations you'll appreciate the necessity of the algebraic arguments, though i will agree hand waving is too dismissive.